Download Extracting bull and bear markets from stock returns

Extracting bull and bear markets from stock returns∗
John M. Maheu† Thomas H. McCurdy‡ Yong Song§
Preliminary
March 2009
Abstract
Bull and bear markets are important concepts used in both industry and
academia. We propose a new Markov-switching model for the identification of
bull and bear regimes for stock returns. Traditional methods to partition the
market index into bull and bear regimes are called dating algorithms. Such an
approach sorts returns ex post based on a deterministic rule. Statistical inference
on returns or investment decisions require more information from the return distribution. Our model fully describes the return distribution while treating bull
and bear regimes as unobservable. The model consists of 4 states, two govern
the bull regime and two govern the bear regime, which allows for rich and heterogeneous intra-regime dynamics. As a result the model can capture bear market
rallies and bull market corrections. A Bayesian estimation approach accounts for
parameter and regime uncertainty and provides probability statements regarding
future regimes and returns. Applied to 123 years of data our model provides superior identification of trends in stock prices. We evaluate both the econometric
specification and the economic value of the model.
∗
The authors are grateful to the Social Sciences and Humanities Research Council of Canada for
financial support.
†
Department of Economics, University of Toronto and RCEA, [email protected]
‡
Rotman School of Management,
University of Toronto and CIRANO, [email protected]
§
Department of Economics, University of Toronto, [email protected]
1
1
Introduction
About the only certainty in the stock market is that, over the long haul, overperformance turns
into underperformance and vice versa (dshort.com, February 22, 2009)
There is a widespread belief both by investors, policy makers and academics that low
frequency trends do exist in the stock market. Traditionally these positive and negative
low frequency trends have been labelled as bull and bear markets respectively. If these
trends do exist, then it is important to extract them from the data to analyse their
properties and consider their use as inputs into investment decisions and risk assessment.
Traditional methods of identifying bull and bear markets are based on an ex post
assessment of the peaks and troughs of the price index. Formal dating algorithms based
on a set of rules for classification are found in Gonzalez, Powell, Shi, and Wilson (2005),
Lunde and Timmermann (2004) and Pagan and Sossounov (2003). Most of this work is
closely related to the dating methods used to identify turning points in the business cycle
(Bry and Boschan (1971)). A significant drawback of this approach is that a turning
point can only be identified several observations after it occurs. The latent nature of bull
and bear markets is ignored and these methods cannot be used for statistical inference
on returns or for investment decisions which require more information from the return
distribution.
For adequate risk management and investment decisions, we need a probability model
for returns and one for which the distribution of returns changes over time. For time
series that tend to be cyclical, for example, due to business cycles or bull and bear stock
markets, a popular model has been a two-state regime switching model in which the
states are latent and the mixing parameters are estimated from the available data. One
popular family is Markov-switching (MS) models for which transitions between states
are governed by a Markov chain.
Our paper investigates whether we can probabilistically identify low frequency trends
(sometimes referred to as primary trends) in a stock market index and also capture the
salient features of each phase of the market. We propose a Markov-switching structure which jointly characterizes the unobservable bull and bear market regimes for stock
returns, allows intra-regime dynamics, and provides a full description of the return distribution. This approach allows uncertainty about the market regime to be incorporated
into out-of-sample forecasts.
Hamilton (1989) applied a two-state MS model to quarterly U.S. GNP growth rates
in order to identify business cycles and estimate 1st-order Markov transition probabilities associated with the expansion and recession phases of those cycles. Durland and
McCurdy (1994) extended this model to allow the transition probabilities to be a function of duration in the state and applied this duration-dependent MS model to business
cycles.
There have been many applications of regime-switching models to stock returns. For
2
example, Hamilton and Lin (1996) relate business cycles and stock market regimes. Cecchetti, Lam, and Mark (2000) and Gordon and St-Amour (2000) derive implications of
regime-switching consumption for equity returns. Maheu and McCurdy (2000) allow
duration-dependent transition probabilities, as well duration-dependent intra-state dynamics for returns and volatilities. Guidolin and Timmermann (2002), Guidolin and
Timmermann (2005), Guidolin and Timmermann (2008), Perez-Quiros and Timmermann (2001) and Turner, Startz, and Nelson (1989), among others, explore the implications of nonlinearities due to regimes switches for asset allocation and/or predictability
of returns.
We allow 4 latent states, two govern the bull regime and two govern the bear regime.
This structure allows for rich and time-varying intra-regime dynamics. In particular,
the model can accommodate short-term reversals (secondary trends) within each regime
of the market. For example, in the bull regime it is possible to have a series of persistent negative returns (a correction), despite the fact that the expected long-run return
(primary trend) is positive in that regime. Bear markets often exhibit persistent rallies
which are subsequently reversed as investors take the opportunity to sell with the result
that the expected long-run return is still negative.
Separating short-term reversals in the market from the primary trend is an important
empirical regularity that a model must capture for it to be able to reproduce the salient
features of the market. This approach is consistent with the definition of bull and bear
markets used by Sperandeo (1990) and Chauvet and Potter (2000). It is also consistent
with the design of the Lunde and Timmermann (2004) filter to capture long-run structure
in stock prices.
Each bull and bear regime has two states. We identify the model by imposing the
long-run mean of returns to be negative in the bear market and positive in the bull
market. We also impose that the overall mean for returns be positive, while allowing
for very different dynamics in each regime. We consider several versions of the model
in which the variance dynamics are decoupled from the mean dynamics. We find that
a model in which the first and second moment are coupled provides the best fit to the
data.
A Bayesian estimation approach accounts for parameter and regime uncertainty and
provides probability statements regarding future regimes and returns. Applied to 123
years of data our model provides superior identification of trends in stock prices.
One important difference with our specification is that the richer dynamics in each
regime allow us to extract bull and bear markets in higher frequency data. As we show,
a problem with a two-state Markov-switching model applied to higher frequency data is
that it results in too many switches between the high and low return states. In other
words, it is incapable of extracting the low frequency trends in the market. In high
frequency data it is important to allow for short-term reversals in the regime of the
3
market.
Our model provides a realistic identification of bull and bear markets and closely
matches the output from traditional dating algorithms. The model also provides a good
fit to the statistics of the cycle. The use of 4 states is important to the success of
our approach. Relative to a two-state model we find that market regimes are more
persistent and there is less erratic switching. According to Bayes factors, our 4-state
model of bull and bear markets is strongly favored over several alternatives including a
two-state model, and different variance dynamics.
Of primary importance is the fact that our model can tell us the probability of a
bull or bear regime in real time, unlike the dating algorithms. It can also produce outof-sample forecasts, something we explore in this paper. We consider several outputs
from the model to perform market timing strategies to assess the economic value of the
trends we extract from the data. In out-of-sample exercises the model provides valuable
probability statements concerning the predictive density of returns. These probability
statements are used to signal long, short and cash positions that allow an investor to
improve on a pure cash position or a buy and hold strategy.
This paper is organized as follows. The next section describes the data, Section 3 discusses existing ex post market dating algorithms. Section 4 summarizes the benchmark
model and develops our proposed specification, and estimation and model comparison
are discussed in Section 5. Section 6 presents results including parameter estimates,
probabilistic identification of bull and bear regimes, and an analysis of the economic
value of our proposed model through market timing strategies and Value-at-Risk forecasts. Section 7 concludes.
2
Data
We begin with 123 years of daily returns (33926 observations). The 1885-1925 daily returns are from Schwert (1990). The 1926-2007 daily returns are the Center for Research
on Security Prices (CRSP) value weighted including distributions (VWRETD) index
returns for the NYSE+AMEX+NASDAQ stock exchanges. We convert daily returns to
continuously compounded returns by taking the natural logarithm of the gross return.
We construct weekly continuously compounded returns from the daily continuously compounded returns by cumulating daily returns from Wednesday close to Wednesday close
of the following week. If a Wednesday is missing, we use Tuesday close. If the Tuesday is also missing, we use Thursday. Weekly returns are scaled by 100 so they are
percentage returns. Unless otherwise indicated, henceforth returns implies continuously
compounded percentage returns. Summary statistics are shown in table 1.
4
3
Bull and Bear Dating Algorithms
Ex post sorting methods for classification of stock returns into bull and bear phases are
called dating algorithms. Such algorithms attempt to use a sequence of rules to isolate
patterns in the data. A popular algorithm is that used by Bry and Boschan (1971) to
identify turning points of business cycles. Pagan and Sossounov (2003) adapted this
algorithm to study the characteristics of bull/bear regimes in monthly stock prices.
First a criterion for identifying potential peaks and troughs is applied; then censoring
rules are used to impose minimum duration constraints on both phases and complete
cycles. Finally, an exception to the rule for the minimum length of a phase is allowed
to accommodate ’sharp movements’ in stock prices.
The Pagan and Sossounov (2003) adaptation of the Bry-Boschan (BB) algorithm can
be summarized as follows:
1. Identify the peaks and troughs by using a window of 8 months.
2. Enforce alternation of phases by deleting the lower of adjacent peaks and the higher
of adjacent troughs.
3. Eliminate phases less than 4 months unless changes exceed 20%.
4. Eliminate cycles less than 16 months.
Window width and phase duration constraints will depend on the particular series and
will obviously be different for smoothed business cycle data than for stock prices. Pagan
and Sossounov (2003) provide a detailed discussion of their choices for these constraints.
There are alternative dating algorithms or filters for identifying turning points. For
example, the Lunde and Timmermann (2004) (LT) algorithm identifies bull and bear
markets using an cumulative return threshold of 20% to locate peaks and troughs moving
forward.1 They define a binary market indicator variable It which takes the value 1 if
the stock market is in a bull state at time t and 0 if it is in a bear state. The stock price
at the end of period t is labelled Pt . Our application of their filter can be summarized
as:
1. Use a 6-month window to locate the initial local maximum or minimum.
2. Suppose we have a local maximum at time t0 , in which case we set Ptmax
= Pt0 .
0
3. Define stopping-time variables associated with a bull market as
τmax (Ptmax
, t0 | It0 = 1) = inf{t0 + τ : Pt0 +τ ≥ Ptmax
}
0
0
τmin (Ptmax
, t0 | It0 = 1) = inf{t0 + τ : Pt0 +τ ≤ 0.8Ptmax
}
0
0
1
Lunde and Timmermann (2004) explore alternative thresholds and also asymmetric thresholds for
switching from bull versus from bear markets. For this description we use a threshold of 20%.
5
4. If τmax < τmin , update the new peak value Ptmax
= Pt0 +τmax and discard the
0 +τmax
previous peak at time t0 . If τmax > τmin , we find a trough at time t0 + τmin . Record
the value Ptmin
= Pt0 +τmin and mark time t0 as one peak
0 +τmin
5. If t0 is a local minimum, bear market stopping times are
}
τmin (Ptmin
, t0 | It0 = 0) = inf{t0 + τ : Pt0 +τ ≤ Ptmin
0
0
τmax (Ptmin
, t0 | It0 = 0) = inf{t0 + τ : Pt0 +τ ≥ 1.2Ptmin
}
0
0
6. If τmin < τmax , update the trough point forward, otherwise it triggers a bull market,
we record the value Ptmax
= Pt0 +τmax and mark time t0 as one trough.
0 +τmax
7. Repeat this process from step 2 until the last data point.
The classification into bull and bear regimes using these two filters is found in Table 2.
There are several features to note. First, the sorting of the data is broadly similar but
with important differences. For example, during the 1930s the BB approach finds many
more switches between the market phases than the LT routine does. More recently, both
identify 1987-10 as a trough but the subsequent bull phase ends in 1998-07 for LT but
2000-03 for BB. Generally, the BB filter identifies more bull and bear markets (31 and
31) than the LT filter (24 and 25). The average bear duration is 53.7 (BB) and 47.2
(LT) weeks while the average bull durations are very different, 152.4 (BB) and 217.0
(LT). In other words, the different parameters and assumptions in the filtering methods
can result in a very different classification of market phases.
Although such dating algorithms can filter the data to locate different regimes, they
cannot be used for forecasting or inference. A two-step approach which involves first
sorting the data into market regimes and then following with an econometric model
conditional on regimes is possible. Such an approach ignores uncertainty from regime
estimation and does not allow it to be incorporated into the second step of estimation
and forecasting. In addition, since the sorting rule focuses on the first moment, it does
not characterize the full distribution of returns. The latter is required if we wish to
derive features of the regimes that are useful for measuring and forecasting risk.
Further, the dating algorithms sort returns into a particular regime with probability
zero or one. However, the data provides more information; investors may be interested
in estimated probabilities associated with the particular regimes. Such information can
be used to answer questions such as ’How likely is it that the market could turn into a
bear next month?’ or ’Are we in a bear market now or just a correction’ ? Probabilistic
modeling of latent states can help answer such questions.
Nevertheless, the dating algorithms are still very useful. For example, we use the
BB and LT algorithms to sort data simulated from our candidate parametric models in
6
order to determine whether the latter can match commonly perceived features of bull
and bear markets.
4
Models
In this section, we briefly review a benchmark two-state model, our proposed 4-state
model, and some alternative specifications of the latter used to evaluate robustness of
our best model.
4.1
Two-State Markov-Switching Model
The concept of bull and bear markets suggests cycles or trends that get reversed. Since
those regimes are not observable, as discussed in Section 1, two-state latent-variable MS
models have been applied to stock market data. A two-state 1st-order Markov model
can be written
rt |st ∼ N (µst , σs2t )
pij = P (st = j|st−1 = i)
(4.1)
(4.2)
i = 0, 1, j = 0, 1. We impose µ0 < 0 and µ1 > 0 so that st = 0 is the bear market and
st = 1 is the bull market.
Modeling of the latent regimes, regime probabilities, and state transition probabilities, allows explicit model estimation and inference. In addition, in contrast to dating
algorithms or filters, forecasts are possible. Investors can base their investment decisions
on the posterior states or the whole forecast density.
4.2
New 4-state Model
Consider the following general K +1 state first-order Markov-switching model for returns
rt |st ∼ N (µst , σs2t )
pij = P (st = j|st−1 = i)
(4.3)
(4.4)
i = 0, ..., K, j = 0, ..., K. We will focus on a 4-state model, K = 3. Without any
additional restrictions we cannot identify the model and relate it to market phases.
Therefore, we consider the following restrictions. First, the states st = 0, 1 are assumed
to govern the bear market; we label these states as the bear regime. The states st = 2, 3
are assumed to govern the bull market; these states are labeled the bull regime. Each
regime has 2 states which allows for positive and negative periods of price growth within
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each regime. In particular
µ0 < 0 (bear negative growth),
(4.5)
µ1 > 0 (bear positive growth),
µ2 < 0 (bull negative growth),
µ3 > 0 (bull positive growth).
This structure can capture short-term reversals in market trends. Each state can have
a different variance and can accommodate autoregressive heteroskedasticity. Therefore,
conditional heteroskedasticity within each regime can be captured.
Consistent with the 2 states in each regime the transition matrix is


p00 p01 0 p03


p10 p11 0
0


P =
0 p22 p23 
0

p30 0 p32 p33
(4.6)
so that each regime will tend to persist and move between positive and negative returns
but can escape to the other regime with probabilties p03 and p30 .2 The unconditional
probabilities associated with P can be solved (Hamilton (1994))
π = (A0 A)−1 A0 e
(4.7)
where A0 = [P 0 − I, ι] and e0 = [0, 0, 0, 0, 1] and ι = [1, 1, 1, 1]0 .
Using the matrix of unconditional state probabilities given by (4.7), we impose the
following conditions on long-run returns in each regime,
π0
π1
µ0 +
µ1 < 0
π0 + π1
π0 + π1
π2
π3
E[rt |bull regime, st = 2, 3] =
µ2 +
µ3 > 0.
π2 + π3
π2 + π3
E[rt |bear regime, st = 0, 1] =
(4.8)
(4.9)
We impose no constraint on the variances.
The equations (4.5) and (4.6), along with equations (4.8) and (4.9), serve to identify
bull and bear regimes.3 The bull (bear) regime has a long-run positive (negative) return.
2
Note that only a negative return bear state can exit to a positive return bull state, and only a
positive return bull state can exit to a negative return bear state. This improves identification of
regimes. It implies, for example, that a bear market rally can never immediately precede a transition
into a bull market. In other words, a bear market rally that turns into a bull market is labelled a bull
market.
3
If the transition probabilities p30 or p03 are sufficienty small, it is possible for st to become trapped
in a bear regime (st = 0, 1) or a bull regime (st = 2, 3). This could result in most, or all, of our data
being sorted into a single regime. To remove this possibility of an absorbing state we impose the bounds
p03 > 0.01, and p30 > 0.01 which ensures a minimum of transitions between bull and bear regimes. The
Gibbs sampling approach to posterior simulation imposes each of these constraints in estimation.
8
Each market regime can display short-term reversals that differ from their long-run
mean. For example, a bear regime can display a bear market rally (temporary period
of positive returns), even though its long-run expected return is negative. Similarly for
the bull market.
4.3
Other Models
Besides the 4-state model we consider several other specifications and provide model
comparisons among them. The dependencies in the variance of returns are the most
dominate feature of the data. This structure may adversely dominate dynamics of the
conditional mean. The following specifications are included to investigate this issue.
4.3.1
Restricted 4-State Model
This is identical to the 4-state model in Section 4.2 except that inside a regime the
return innovations are homoskedastic. That is, σ02 = σ12 and σ22 = σ32 . In this case, the
variance within each regime is restricted to be constant although the overall variance of
returns can change over time due to switches between regimes.
4.3.2
Markov-Switching Mean and i.i.d. Variance Model
In this model the mean and variance dynamics are decoupled. This is a robustness
check to determine to what extent the variance dynamics might be driving the regime
transitions. This specification is identical to the Markov-switching model in Section 4.2
except that only the conditional mean follows the Markov chain while the variance
follows an independent i.i.d mixture. That is,
rt |st = µst + zt
L
∑
zt ∼
ηi N(0, σi2 )
(4.10)
(4.11)
i=1
pij = P (st = j|st−1 = i)
(4.12)
∑
2
is imposed
i, j = 0, ..., K, Li=1 ηi = 1 and ηi ≥ 0. For identification, σ12 < σ22 < · · · < σK
along with the constraints used for the conditional mean in the previous section.
5
5.1
Estimation and Model Comparison
Estimation
In this section we discuss Bayesian estimation for the most general model introduced in
Section 4.2 assuming there are K + 1 total states, k = 0, ..., K. The other models are
9
estimated in a similar way with minor modifications.
2
There are 3 groups of parameters M = {µ0 , ..., µK }, Σ = {σ02 , ..., σK
}, and the elements of the transition matrix P . Let θ = {M, Σ, P } and given data IT = {r1 , ..., rT }
we augment the parameter space to include the states S = {s1 , ..., sT } so that we
sample from the full posterior P (θ, S|IT ). Assuming conditionally conjugate priors
µi ∼ N (mi , n2i ), σi−2 ∼ G(vi /2, si /2) and each row of P following a Dirichlet distribution, allows for a Gibbs sampling approach following Chib (1996). Gibbs sampling
iterates on sampling from the following conditional densities given startup parameter
values for M , Σ and P :
• S|M, Σ, P
• M |Σ, P, S
• Σ|M, P, S
• P |M, Σ, S
Sequentially sampling from each of these conditional densities results in one iteration
of the Gibbs sampler. Dropping an initial set of draws to remove any dependence from
startup values, the remaining draws {S (j) , M (j) , Σ(j) , P (j) }N
j=1 are collected to estimate
features of the posterior density. Simulation consistent estimates can be obtained as
sample averages of the draws. For example, the posterior mean of the state dependent
mean and standard deviation of returns are estimated as
N
1 ∑ (j)
µ ,
N j=1 k
N
1 ∑ (j)
σ ,
N j=1 k
(5.1)
for k = 0, ..., K and are simulation consistent estimates of E[µk |IT ] and E[σk |IT ] respectively.
The first sampling step of S|M, Σ, P involves a joint draw of all the states. Chib
(1996) shows that this can be done by a so-called forward and backward smoother
through the identity
p(S|θ, IT ) = p(sT |θ, IT )
T
−1
∏
p(st |st+1 , θ, It ).
(5.2)
t=1
The forward pass is to compute the Hamilton (1989) filter for t = 1, ..., T
p(st = k|θ, It−1 ) =
K
∑
p(st−1 = l|θ, It−1 )plk , k = 0, ..., K,
(5.3)
l=0
p(st = k|θ, It−1 )f (rt |It−1 , st = k)
, k = 0, ..., K. (5.4)
p(st = k|θ, It ) = ∑K
p(s
=
l|θ,
I
)f
(r
|I
,
s
=
l)
t
t−1
t
t−1
t
l=0
10
Note that f (rt |It−1 , st = k) is the normal pdf N (µk , σk2 ). Finally, Chib (1996) has shown
that a joint draw of the states can be taken sequentially from
p(st |st+1 , θ, It ) ∝ p(st |θ, It )p(st+1 |st , P ),
(5.5)
where the first term on the right-hand side is from (5.4) and the second term is from
the transition matrix. This is the backward step and runs from t = T − 1, T − 2, ..., 1.
The draw of sT is taken according to p(sT = k|θ, IT ), k = 0, ..., K.
The second and third sampling steps are straightforward and use results from the
linear regression model. Conditional on S we select the data in regime k and let the
number of observations of st = k be denoted as Tk . Then µk |Σ, P, S ∼ N (ak , Ak ),

ak = Ak σk−2
∑

−1
 , Ak = (σk−2 Tk + n−2
rt + n−2
k mk
k ) .
(5.6)
t∈{t|st =k}
A draw of the variance is taken from


σk−2 |M, P, S ∼ G (Tk + vk )/2, 
∑


(rt − µk )2 + sk  /2
(5.7)
t∈{t|st =k}
Given the conjugate Dirichlet prior on each row of P , the final step is to sample
P |M, Σ, S from the Dirichlet distribution (Geweke (2005)).
An important byproduct of Gibbs sampling is an estimate of the smoothed state
probabilties P (st |IT ) which can be estimated as
N
1 ∑
p(s\
=
i|I
)
=
1st =i (S (j) )
t
T
N j=1
(5.8)
for i = 0, ..., K.
At each step, if a parameter draw violates any of the prior restrictions in (4.5), (4.6),
(4.9) and (4.8), then it is discarded. For the 4-state model we set the independent priors
as
µ0 ∼ N (−2, 2)1µ0 <0 , µ1 ∼ N (1.5, 2)1µ1 >0 ,
(5.9)
µ2 ∼ N (−1.5, 2)1µ2 <0 , µ3 ∼ N (2, 2)1µ3 >0 ,
(5.10)
(p00 , p01 , p03 ), (p33 , p32 , p30 ) ∼ Dirichlet(0.6, 0.37, 0.03),
(5.11)
(p10 , p11 ), (p23 , p22 ) ∼ Dirichlet(0.4, 0.6), σi−2 ∼ G(1/20, 1/2).
(5.12)
These priors are informative but cover a wide range of empirically relevant parameter
values.
11
5.2
Model Comparison
If the marginal likelihood can be computed for a model it is possible to compare models based on Bayes factors. Non-nested models can be compared as well as specifications with a different number of states. Note that the Bayes factor penalizes
over-parameterized models that do not deliver improved predictions.4 For the general
Markov-switching model with K + 1 states, the marginal likelihood for model Mi is
defined as
∫
(5.13)
p(r|Mi ) = p(r|Mi , θ)p(θ|Mi )dθ
which integrates out parameter uncertainty. p(θ|Mi ) is the prior and
p(r|Mi , θ) =
T
∏
f (rt |It−1 , θ)
(5.14)
t=1
is the likelihood which has S integrated out according to
f (rt |It−1 , θ) =
K
∑
f (rt |It−1 , θ, st = k)p(st = k|θ, It−1 ).
(5.15)
k=0
The term p(st = k|θ, It−1 ) is available from the Hamilton filter. Chib (1995) shows how
to estimate the marginal likelihood for MS models. His estimate is based on re-arranging
Bayes’ theorem as
∗
∗
\i ) = p(r|Mi , θ )p(θ |Mi )
p(r|M
p(θ∗ |r, Mi )
(5.16)
where θ∗ is a point of high mass in the posterior pdf. The terms in the numerator are
directly available above while the denominator can be estimated using additional Gibbs
sampling runs.5
A log-Bayes factor between model Mi and Mj is defined as
log(BFij ) = log(p(r|Mi )) − log(p(r|Mj )).
(5.17)
Kass and Raftery (1995) suggest interpreting the evidence for Mi versus Mj as: not
worth more than a bare mention for 0 ≤ log(BFij ) < 1; positive for 1 ≤ log(BFij ) < 3;
strong for 3 ≤ log(BFij ) < 5; and very strong for log(BFij ) ≥ 5.
4
This is referred to as an Ockham’s razor effect. See Kass and Raftery (1995) for a discussion on
the benefits of Bayes factors.
5
The integrating constant in the prior pdf is estimated by simulation.
12
6
Results
6.1
Parameter Estimates and Implied Distributions
Model estimates for the 2-state Markov-switching model are found in Table 3. This
specification displays a negative conditional mean along with a high conditional variance
and a high conditional mean with a low conditional variance. Both regimes are highly
persistent. These results are consistent with the sorting of bull and bear regimes in
Maheu and McCurdy (2000) and Guidolin and Timmermann (2005).
Estimates for our new 4-state model are found in Table 4. Recall that states st = 0, 1
capture the bear regime while states st = 2, 3 capture the bull regime. Each regime
contains a state with a positive and a negative conditional mean. Consistent with the 2state model, volatility is highest in the bear regime. In particular, the highest volatility
occurs in the bear regime in state 0. This state also delivers the lowest expected return.
The highest expected return and lowest volatility is in state 3 which is part of the bull
regime.
All states are persistent with estimates of pii ranging from 0.68 to 0.97. Compared
to the bear regime, there are more frequent switches inside the bull regime, since the
probabilities p22 and p33 are smaller than p00 and p11 .
The unconditional distribution of the states is reported in Table 5. The probability of
the bear regime is π0 +π1 = 0.511 while the probability of a bull regime is π2 +π3 = 0.489.6
The last column of Table 6 reports posterior quantities associated with the bull and
bear market regimes. The weekly conditional means for the bear and bull regimes are
−0.03% and 0.27% respectively. The intra-regime returns for this model will exhibit
conditional heteroskedasticity. Estimates of the unconditional standard deviations are
3.03 (bear) and 1.42 (bull).
Table 6 also allows a comparison of some regime statistics for the 2-state and 4state models. For example, the expected duration of regimes is longer in the 4-state
model. That is, by allowing heterogeneity within a regime in our 4-state model, we
switch between bull and bear markets less frequently.
In the 2-state model, the expected return and variance are fixed within a regime.
In this case, the only source of regime variance is return innovations. In contrast, the
average variance for each regime in the 4-state model can be attributed to changes in the
conditional mean as well as to the average conditional variance of the return innovations.
For instance, the average variance of returns in the bear regime can be decomposed as
Var(rt |st = 0, 1) = Var(E[rt |st ]|st = 0, 1) + E[Var(rt |st )|st = 0, 1], with a similar result
for the bull regime. For the bear phase, the mean dynamics account for a small share
6
In the following discussion, posterior quantities are computed using the average of the Gibbs sampling draws. In general this will differ from the results derived from the posterior mean of the model
parameters.
13
2% of the total variance, while for the bull it is larger at 11%.7
The MS-2 model has identical higher order moments in each market. However, the
MS-4 specification has very different skewness and kurtosis. For instance, the bear
market displays positive skewness and high kurtosis while the bull market has negative
skewness and relatively lower kurtosis.8 The reason that skewness is positive in the bear
market is that we expect a negative trend in stock prices, but it is the occasional bear
rally that provides higher than normal returns that cause an asymmetry in the right tail
of the regime distribution. The result is exactly the opposite in the bull market which
has a relatively thicker left tail and gives rise to negative skewness.
The expected future return, conditional on starting in each of the four states, is
shown in Figure 1. This Figure plots, for each state i = 0., , , .3, expected future weekly
return, E[rt+h |st = i], for a range of weeks h, that is, for forecast horizons t + h. The
term structure of expected returns in each state can differ significantly, but for a long
enough forecast horizon they converge to the unconditional mean of returns.
Figures 2 and 3 provide more details on the distributional features of each state. The
first figure is the predictive densities in each of the 4 states. All states have either different
location and/or different tail shapes. Integrating the 2 states in each regime, generates
the implied predictive densities for the bull and bear market illustrated in Figure 3. Also
included is the implied unconditional distribution of returns. As mentioned above, the
bull regime has a positive mean and more mass around 0 relative to the bear distribution
which has much thicker tails and a negative mean. The unconditional distribution is a
mixture of these 2 regime densities.
An important feature of the 4-state model relative to the 2-state version is that the
4-state model allows the realized conditional mean associated with a particular regime to
change over time. This is because a regime can have a different sequence of states realized
during different historical periods. As an example, consider Figure 4. Data from the 4state model is simulated; the top panel displays the cumulative return, the second panel
the realized state, and the bottom panel the average conditional return in a bull or bear
regime.9 The figure shows that the realized conditional mean assoicated with a particular
regime will be different over time. In addition, the returns will display heteroskedasticity
inside a regime. These are features that the simpler 2-state parameterization cannot
capture. In that model all bull (bear) markets have an identical conditional mean and
conditional variance.
7
This is computed as 0.22/(0.22+8.98) and 0.22/(0.22+1.74).
The implied unconditional skewness is -0.11 and kurtosis 7.14.
9
For instance, the average conditional return for the sequence of bull states in {1, 0, 3, 2, 3, 3, 0} is
1
3
4 µ2 + 4 µ3 .
8
14
6.2
Model Comparisons
One can conduct formal model comparisons based on the marginal likelihoods reported
in Table 7. The constant mean and variance model performs the worst. The next model
has a constant mean but allows the variances to follow a 4-state i.i.d. mixture. Following
this are models with a 2-state versus a 4-state Markov-switching conditional mean – both
combined with a 4-state i.i.d. variance as in Section 4.3.2. In both cases, the additional
dynamics that are introduced to the conditional mean of returns provides a significant
improvement. However, these specifications are strongly dominated by their counterparts which allow a common 2 (and 4) state Markov chain to direct both conditional
moments. These specifications capture persistence in the conditional variance.
Note that the log-Bayes factor between the 2-state MS and the 4-state MS in the
conditional mean restricted to have only a 2-state conditional variance (Section 4.3.1)
is large at 41.7 = −13386.8 − (−13428.5). This improved fit comes when additional
conditional mean dynamics (going from 2 to 4 states) are added to the basic 2-state MS
model. The best model is the 4-state Markov-switching model. The log-Bayes factor
in support of the 4-state versus the 2-state model is 107.7 = −13320.8 − (−13428.5).
This is very strong evidence that the 4-state specification provides a better fit to weekly
returns.
The Markov-switching models specify a latent variable that directs low frequency
trends in the data. As such, the regime characteristics from the population model are
not directly comparable to the dating algorithms of Section 3. Instead we consider the
dating algorithm as a lens to view both the CRSP data and data simulated from our
models. Using parameter draws from the Gibbs sampler, we simulate return data from
a model and then apply both the BB and the LT dating algorithm to those simulated
returns. This is done many times10 and the average and 0.70 density intervals of these
statistics are reported in Tables 8 and 9. The tables also include the statistics from the
CRSP data and a ∗ indicates that a model’s 0.70 density interval does not contain the
CRSP statistic.
Based on these results, the 2-state model is unable to account for 6 of the data
statistics while the 4-state model cannot account for 4 for the BB dating approach. It is
not surprising that the 2-state model fails to capture the intra-regime dynamics in the
second panel of Table 8 as this is what the 4-state model is designed to do. Based on
the LT dating algorithm both models are evenly ranked with 5 statistics that are not
contained in the density intervals. We conclude that the 4-state model generally does
as well and often better than the 2 state model, nevertheless, there is some room for
improvement. The average bear negative return duration and the average bull positive
return duration are difficult for the models to match, producing values too low relative
to the data. According to the LT dating approach, the 4-state model’s bull duration
10
10,000 simulations each of 6389 observations.
15
and standard deviation are too low.
The dating of the market regimes using the LT approach are found in the top panel
of Figure 5. The shaded portions under the cumulative return denote bull markets while
the white portions of the figure are the bear markets. Below this panel is the smoothed
probability of a bull market, P (st = 2|IT ) + P (st = 3|IT ) for the 4-state model. The
final plot in Figure 5 is the smoothed probability of a bull market, P (st = 1|IT ) from the
2-state model. The 4-state model produces less erratic shifts between market regimes,
closely matches the trends in prices, and generally corresponds to the dating algorithm.
The two-state model is less able to extract the low frequency trends in the market. In
high frequency data it is important to allow intra-regime dynamics, such as short-term
reversals.
Note that the success of our model should not be based on how well it matches
the results from dating algorithms. Rather this comparison is done to show that the
latent-state MS models can identify bull and bear markets with similar features to those
identified by conventional dating algorithms. Beyond that, the Markov-switching models
presented in this paper provide a superior approach to modeling stock market trends as
they deliver a full specification of returns along with latent market dynamics. Such an
approach permits out-of-sample forecasting which we turn to next.
6.3
Market Timing
To investigate the value of the model in its ability to identify and predict trends in
stock returns, we consider some simple market timing strategies based on the predictive
density.
The predictive density for future returns based on current information at time t is
computed as
∫
p(rt |It−1 ) =
f (rt |θ, It−1 )p(θ|It−1 )dθ
(6.1)
which involved integrating out both state and parameter uncertainty using the posterior
distribution p(θ|It−1 ). From the Gibbs sampling draws {S (j) , M (j) , Σ(j) , P (j) }N
j=1 based
on data It−1 we approximate the predictive density as
N
K
1 ∑∑
(i)
\
p(rt |It−1 ) =
f (rt |θ(i) , It−1 , st = k)p(st = k|st−1 , θ(i) )
N i=1 k=0
(i)
2(i)
(i)
(6.2)
where f (rt |θ(i) , It−1 , st = k) follows N (µk , σk ) and p(st = k|st−1 , θ(i) ) is the transition
probability.
Consider an investor with wealth Wt−1 who has the option to invest in a risk-free
asset yielding rf or to take a long or short position in the market. The investor takes a
16
long position, using the proportion αL of wealth, if the model indicates that
P (rt > rf |It−1 ) > CVL
(6.3)
where CVL is a specified probability threshold. Next period’s wealth will be
Wt = αL Wt−1
Pt
+ (1 − αL )(1 + rf )Wt−1
Pt−1
(6.4)
Analogously, the investor takes a short position, using a proportion αS of wealth if
P (rt < 0|It−1 ) > CVS
(6.5)
where CVL is a specified probability threshold. In this case the next period wealth is
(
Wt = αS Wt−1
Pt
(1 + rf ) −
Pt−1
)
+ (1 + rf )Wt−1 .
(6.6)
Otherwise, the investor puts all his or her wealth in the risk-free asset, in which case
wealth next period will be Wt = (1 + rf )Wt−1 .
We investigate the challenging out-of-sample period of 2008. At each point in the
sample we re-estimate the model and compute the one-week-ahead predictive density
and the associated probabilities required for the above investment strategy. The weekly
risk-free rate was obtained from the website of Kenneth French. Given an initial wealth
of $100, Table 10 reports wealth at the end of 2008 for various values of αL , CVL , αS , CVS .
The bottom panel of the table reports final wealth from leaving all wealth in the riskfree asset every period; as well as from buying and holding the market index for the full
investment horizon.
The buy and hold strategy performs particularly poorly during this period, resulting
in a loss of 40% of initial wealth; all other strategies result in better outcomes. The
most profitable approach allows for short positions to be taken. For example, taking
short positions with 80% of current wealth and long positions of 20% provides the best
performance (Sharpe ratio of 0.088) amongst those strategies that we implemented.
This suggests that the model can provide effective signals concerning the direction of
the market.
In addition, we forecast out-of-sample returns using the predictive mean versus the
sample average. The model (predictive mean) achieves a mean-squared error of 17.28
versus 17.99 for the sample mean. This provides further evidence that the model is
capturing trends in the data.
17
6.4
Value-at-Risk
An industry standard measure of potential portfolio loss is the Value-at-Risk (VaR).
VaR(α),t is defined as the 100α percent quantile of the portfolio value or return distribution given information at time t − 1. We compute VaR(α),t from the predictive density
MS-4 model as
P (rt < VaR(α),t |It−1 ) = α.
(6.7)
Given a correctly specified model the probability of a return of VaR(α),t or less is α.
To compute the Value-at-Risk from the MS-4 model we do the following. First, N
draws from the predictive density are taken as follows: draw θ and st−1 from the Gibbs
sampler, a future state s̃t is simulated based on P and r̃t |s̃t ∼ N (µs̃t , σs̃2t ). From the
resulting draws, the r̃t with rank [N α] is an estimate of VaR(α),t .
Figure 6 displays the conditional VaR from 2005 – 2008 predicted by the MS-4 model,
as well as that implied by the normal benchmark for α = 0.05. At each point the model
is estimated based on information up to t − 1. Similarly, the bechmark, N (0, s2 ), sets s2
to the sample variance using It−1 . It is clear that the normal benchmark overestimates
the VaR for much of the sample and then tends to understate it in 2007-2008. The MS-4
has a very different VaR(.05),t over time becasue it takes into account the current regime.
The potential losses increase considerably after 2007 as the model identifies a move from
a bull to a bear market.
7
Conclusion
This paper proposes a new 4-state Markov-switching model to identify bull and bear
markets in weekly stock market data. The model fully describes the return distribution
while treating bull and bear regimes as unobservable. Of the 4 latent states, two govern
the bull regime and two govern the bear regime. This allows for rich and heterogeneous
intra-regime dynamics.
The model provides a realistic identification of bull and bear markets and closely
matches the output from traditional ex post dating algorithms. The model also provides
a good fit to the statistics of the cycle. Relative to a two-state model we find that market
regimes are more persistent and there is less erratic switching. Model comparisons show
that the 4-state specification of bull and bear markets is strongly favored over several
alternatives including a two-state model, as well as various alternative specifications for
variance dynamics.
In out-of-sample exercises the model provides probability statements concerning the
predictive density of returns. These probability statements are used to signal long, short
and cash positions that allow an investor to improve on a pure cash position or a buy
18
and hold strategy.
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20
Table 1: Weekly Return Statistics (1885-2007)a
a
b
N
Mean
standard deviation
Skewness
Kurtosis
J-Bb
6389
0.173
2.290
-0.50
7.3
5178∗
Continuously compounded returns
Jarque-Bera normality test: p-value = 0.00000
Table 2: BB and LT Dating Algorithm Turning Points
Troughs
BB
LTb
a
Peaks
BB
Troughs
BB
LT
LT
1885-04
1887-10
1890-12
1893-07
1896-08
1900-06
1903-10
1907-11
1910-07
1914-12
1917-12
1921-06
1932-06
1935-03
a
b
1890-12
1893-07
1896-08
1903-10
1907-11
1917-12
1921-06
1929-11
1930-12
1932-06
1933-03
1933-10
1887-05
1890-06
1892-03
1895-09
1899-09
1902-09
1906-10
1909-09
1912-10
1916-11
1919-11
1929-09
1934-02
1937-03
1938-03
1940-05
1942-04
1947-05
1949-06
1953-09
1957-11
1960-09
1962-06
1966-10
1970-05
1971-11
1974-10
1982-08
1984-07
1987-10
1890-06
1892-03
1895-09
1902-09
1906-10
1916-11
1919-11
1929-09
1930-04
1931-02
1932-09
1933-07
2002-10
1938-03
1940-05
1942-04
1947-05
1962-06
1970-05
1974-10
1982-08
1987-10
1998-10
2002-10
Peaks
BB
LT
1939-10
1941-09
1946-05
1948-06
1953-03
1957-07
1960-01
1961-12
1966-02
1968-12
1971-04
1973-01
1980-11
1983-06
1987-08
1939-10
1941-09
1946-05
2000-03
2007-10
1961-12
1968-12
1973-01
1980-11
1987-08
1998-07
2000-03
2007-10
1937-03
BB: Bry and Boschan algorithm using Pagan and Sossounov parameters
LT: Lunde and Timmermann algorithm
21
Table 3: MS Two-State Model Estimates
µ0
µ1
σ0
σ1
p00
p11
Mean
Median
Std
0.95 DI
-0.42
0.31
4.12
1.57
0.94
0.99
-0.42
0.31
4.12
1.57
0.94
0.99
0.13
0.02
0.12
0.02
0.01
0.002
(-0.68, -0.18)
( 0.26, 0.35)
( 3.88, 4.38)
( 1.52, 1.62)
( 0.92, 0.96)
( 0.98, 0.99)
This table reports the posterior mean,
median, standard deviation and 0.95 density
intervals for model parameters.
Table 4: MS 4-State Model Estimates
µ0
µ1
µ2
µ3
σ0
σ1
σ2
σ3
p00
p01
p03
p10
p11
p22
p23
p30
p32
p33
Mean
Median
Std
0.95 DI
-0.81
0.26
-0.41
0.69
4.72
1.99
1.64
1.06
0.89
0.08
0.03
0.03
0.97
0.68
0.32
0.01
0.20
0.78
-0.80
0.26
-0.37
0.69
4.72
1.99
1.88
1.06
0.90
0.09
0.03
0.03
0.97
0.69
0.31
0.01
0.20
0.79
0.149
0.047
0.198
0.065
0.171
0.062
0.089
0.053
0.021
0.017
0.008
0.006
0.006
0.082
0.082
0.002
0.047
0.048
(-1.109, -0.528)
( 0.164, 0.346)
(-0.950, -0.134)
( 0.565, 0.817)
( 4.410, 5.080)
( 1.883, 2.126)
( 1.439, 1.802)
( 0.946, 1.155)
( 0.849, 0.929)
( 0.050, 0.117)
( 0.013, 0.045)
( 0.019, 0.043)
( 0.957, 0.981)
( 0.483, 0.815)
( 0.185, 0.517)
( 0.010, 0.018)
( 0.120, 0.302)
( 0.683, 0.868)
The posterior mean, median, standard
deviation and 0.95 density intervals for model
parameters.
22
Table 5: Unconditional State Probabilites
mean
π0
π1
π2
π3
0.139
0.372
0.207
0.282
0.95 DI
(0.096,
(0.241,
(0.110,
(0.185,
0.195)
0.510)
0.322)
0.390)
The posterior mean and 0.95 density intervals
associated with the posterior distribution for π
from Equation (4.7).
Table 6: Implied Regime Statistics for MS Models
bear mean
bear duration
bear standard deviation
bear variance from Var(E[rt |st ]|st = 0, 1)
bear variance from E[Var(rt |st )|st = 0, 1]
MS-2
MS-4
-0.42
(-0.68, -0.18)
16.5
(11.9, 22.7)
4.12
(3.89, 4.38)
0.00
-0.03
(-0.108, -0.001)
153.9
(78.8, 288.7)
3.03
(2.81, 3.28)
0.22
(0.10, 0.38)
8.98
(7.70, 10.5)
0.07
(0.01, 0.15)
5.25
(4.74,5.81)
bear skewness
17.0
(15.1, 19.2)
0
bear kurtosus
3
bull mean
bull duration
bull standard deviation
bull variance from Var(E[rt |st ]|st = 2, 3)
bull variance from E[Var(rt |st )|st = 2, 3]
0.31
(0.26, 0.35)
71.0
(50.6, 100.0)
1.57
(1.52, 1.62)
0.00
bull skewness
2.47
(2.32, 2.63)
0
bull kurtosus
3
0.27
(0.21, 0.33)
136.6
(94.4, 180.3)
1.42
(1.36, 1.49)
0.22
(0.11, 0.36)
1.74
(1.45, 2.00)
-0.66
(-0.85,-0.45)
3.075
(2.28,3.60)
The posterior mean and 0.95 density interval for regime statistics.
23
Table 7: Log Marginal Likelihoods: Alternative Models
log f (Y | Model)
Model
Constant mean with constant variance
Constant mean with 4-state i.i.d variance
MS 2-state mean with 4-state i.i.d. variance (4.10 with K + 1 = 2)
MS 4-state mean with 4-state i.i.d. variance (4.10 with K + 1 = 4)
MS 2-state mean with coupled MS 2-state variance (4.1)
MS 4-state mean with coupled MS 4-state variance (4.3)
MS 4-state mean with coupled MS 2-state variance (σ02 = σ12 , σ22 = σ32 )
24
-14425.1
-13808.6
-13698.6
-13580.4
-13428.5
-13320.8
-13386.8
Table 8: Posterior bull/bear statistics by BB algorithm
Avg. number of bears
CRSP
MS-2
MS-4
31
31.4
(28, 35)
49.9
(42.5, 57.2)
-33.0
(-38.5, -27.6)
-0.67
(-0.80, -0.55)
2.71*
(2.50, 2.92)
31.5
(28, 35)
155.9
(130.3, 181.9)
68.5
(57.6, 79.6)
0.44
(0.41, 0.47)
2.00
(1.89, 2.11)
34.1
(30, 38)
55.4
(47.6, 63.2)
-36.9
(-42.7, -31.1)
-0.67
(-0.78, -0.57)
2.75*
(2.51, 2.98)
34.2
(30, 38)
134.1
(113.1, 155.2)
59.1*
(50.8, 67.5)
0.44
(0.41, 0.47)
2.03
(1.89, 2.17)
-2.37
(-2.58, -2.16)
1.72
(1.57, 1.87)
2.26*
(2.17, 2.36)
1.74*
(1.66, 1.82)
-1.35*
(-1.41, -1.29)
1.63*
(1.57, 1.70)
1.66
(1.63, 1.69)
2.43*
(2.37, 2.49)
-2.42
(-2.63, -2.22)
1.68
(1.54, 1.83)
2.26*
(2.18, 2.34)
1.81
(1.74, 1.89)
-1.41
(-1.50, -1.33)
1.59
(1.48, 1.69)
1.65
(1.62, 1.69)
2.62*
(2.52, 2.71)
Avg. bear duration
53.7
Avg. bear amplitudeb
-34.4
Avg. bear returnc
-0.64
Avg. bear standard deviationd
2.44
Avg. number of bulls
31
Avg. bull duration
152.4
Avg. bull amplitude
70.0
Avg. bull return
0.46
Avg. bull standard deviation
1.95
Avg. bear -ve return
-2.27
Avg. bear +ve return
1.65
Avg. bear -ve return duration
2.39
Avg. bear +ve return duration
1.84
Avg. bull -ve return
-1.41
Avg. bull +ve return
1.54
Avg. bull -ve return duration
1.65
Avg. bull +ve return duration
2.77
a
70% density interval
Aggregate return over one regime
c
Average return in one regime
d
Return standard deviation in one regime
b
25
Table 9: Posterior bull/bear statistics by LT Algorithm
Avg. number of bears
CRSP
MS-2
MS-4
25
26.8
(21, 32)
41.7
(34.0, 49.3)
-39.1
(-43.6, -34.6)
-0.96
(-1.11, -0.81)
3.10*
(2.91, 3.29)
26.4
(21, 32)
209.2
(158.5, 261.4)
83.5
(66.4, 100.8)
0.40
(0.37, 0.44)
2.14*
(1.99, 2.30)
30.6
(25, 37)
48.0
(39.8, 56.3)
-42.3
(-46.9, -37.7)
-0.90
(-1.04, -0.76)
3.17
(2.95, 3.38)
30.3
(24, 36)
170.5*
(128.5, 212.7)
68.9*
(56.2, 81.6)
0.41
(0.37, 0.46)
2.22*
(2.05, 2.39)
-2.72
(-2.94, -2.50)
1.83
(1.67, 1.98)
2.37
(2.26, 2.48)
1.66*
(1.58, 1.73)
-1.36
(-1.41, -1.30)
1.63*
(1.57, 1.69)
1.69
(1.66, 1.73)
2.39*
(2.33, 2.44)
-2.69
(-2.91, -2.48)
1.75
(1.61, 1.89)
2.34*
(2.24, 2.43)
1.73
(1.68, 1.80)
-1.41
(-1.49, -1.34)
1.58
(1.48, 1.69)
1.69
(1.65, 1.72)
2.56*
(2.48, 2.65)
Avg. bear duration
47.2
Avg. bear amplitudeb
-40.5
Avg. bear returnc
-0.86
Avg. bear stdd
3.34
Avg. number of bulls
24
Avg. bull duration
217.0
Avg. bull amplitude
88.2
Avg. bull return
0.41
Avg. bull std
2.41
Avg. bear -ve return
-2.70
Avg. bear +ve return
1.86
Avg. bear -ve return duration
2.43
Avg. bear +ve return duration
1.79
Avg. bull -ve return
-1.37
Avg. bull +ve return
1.52
Avg. bull -ve return duration
1.72
Avg. bull +ve return duration
2.69
a
70% density interval
Aggregate return over one regime
c
Average return in one regime
d
Return standard deviation in one regime
b
26
Table 10: Market
αL CVL
0.2 0.5
0.5 0.5
0.5 0.5
1.0 0.5
0.5 0.5
0.5 0.5
0.2 0.5
Buy&Hold 1.0
Cash 0.0
Timing Out-of-Sample (2008)
αS CVS
WT Sharpe ratio
0.2 0.5 102.32
0.011
0.5 0.5 102.37
0.005
0.0 1.0
91.61
-0.208
0.0 1.0
81.96
-0.212
0.6 0.5 104.47
0.021
0.8 0.5 108.61
0.042
0.8 0.5 115.81
0.088
0.0
59.99
-0.244
0.0
101.86
27
0.6
S=0
S=1
S=2
S=3
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
10
20
30
40
50
60
70
80
90
100
Figure 1: MS 4 States, Conditional Mean Dynamics: E[rt+h |st = S] versus h.
28
0.4
S=0
S=1
S=2
S=3
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-10
-5
0
5
10
Figure 2: MS 4 States, State Density
0.35
Bear
Bull
Unconditional
0.3
0.25
0.2
0.15
0.1
0.05
0
-10
-5
0
5
Figure 3: MS 4 States, Regime Density
29
10
Figure 4: Simulated Returns, States and Average Regime Conditional Mean
30
-1
-0.7
-0.4
-0.1
0.2
0
1
2
0
3
20
40
60
80
100
120
140
0
100
200
Conditional mean
Simulated states
300
400
Simulated cumulative log return * 100
500
600
700
800
900
1000
Figure 5: Cumulative Return (LT dating) & Smoothed Bull States from MS-4 and MS-2
31
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
0
1
200
400
600
800
1000
1200
1892
1904
1916
1928
1940
1952
1964
1976
1988
cumulative log return * 100
2000
Figure 6: Value-at-Risk from MS-4 and Benchmark Normal distribution
32
200501
0
0.2
0.4
0.6
0.8
-20
1
-15
-10
-5
0
5
10
200507
200601
200607
200701
200707
200801
200807
Filtered Probability of Bull
return
Model
Normal